Proposition 1.1.1 (Vinnikov, [1]})
There exists a canonical homology
basis \(\lbrace a_j,b_j\rbrace _{j=1}^n\)  ("Vinnikov basis") on \(\mathcal {K}_n\)  such that
the action of \(\tau \)  on the homology group \(\textrm {H}_1(\mathcal {K}_n,\mathbb {Z})\) 
is
\(&\left(\begin{array}{c}\tau (\underline{a}) \\\tau (\underline{b}) \\\end{array}\right)=\left(\begin{array}{cc}I_n & 0 \\H & -I_n \\\end{array}\right)\left(\begin{array}{c}\underline{a} \\\underline{b} \\\end{array}\right),\) 
